One of the biggest
obstacles to overcome in the quest for accurate measurements is
excess noise and interference that obstructs the signal from your
sample. In other areas of this site, there are explanations of the
many steps that can be taken during the experiment to reduce noise.
Note, however, that the connections between the magnet and data
acquisition devices are arranged to help reduce as much noise as
possible. A central component of this arrangement is the Lock-In
Amplifier.

Lock-In Amplifiers
detect the signal and any noise associated with it (including thermal,
shot and 1/f noise). Because of this, it is necessary to take noise
into consideration when choosing a frequency for the measurement.
There is substantial 1/f noise at zero frequency, making DC undesirable.
Measuring at a higher frequency usually results in a less noisy
environment. However, if you go too high in frequency, cable capacitance
and inductance may obscure the signal. Therefore, it is best to
test your system at chosen frequencies to determine the appropriate
one for your particular sample and measurement.

Taking an average
of the signal is the first way to eliminate noise and isolate the
signal. Averaging over a time t enables you to measure noise over
a bandwidth ~1/t, no matter what the excitation frequency is. The
noise you will
see is just

Where
is the noise density at that frequency.

Once an excitation frequency is chosen, a mixer is used to isolate
the signal. A mixer is an electronic gadget that multiplies, instantaneously,
two voltages. The two common modes of operation of the mixer and
related signals and excitation frequency are homodyne detection
and heterodyne detection.

Heterodyne detection
is when . This
is better.
Homodyne detection is what lock-ins use and is when .
If then

then

The mixer output has components at 2f. Also, at DC the amplitude
of the DC component is dependent on the phase of the signal relative
to the reference, and the size of the signal.

It is important
to know that the averaging time determines the bandwidth. This is
crucial if the signal is changing. For example, if the signal varies
at 100 Hz, then the averaging time has to be less than 10ms.

Conversely,
if there is an interference source at 60 Hz, and you want to measure
at 200 Hz, while the signal varies at 10 Hz, then: